How to tell a neural network that: "your i-th input is special"
Assume the image can contain objects of class $C_1 \dots C_c$. Assume a set of additional inputs that has a meaning of questions as &quot;contains the image a C_i or C_j or ... ?&quot;.
The main problem for the system is classify the image in classes $C_i$. Second problem is answer the implicit question proposed by the remainder inputs.
Thus, better combine two NNs:

first one an object recognizer, with input the image data.
second one a NN that will answer the question implicit in remainder bits, with inputs the output of previous NN and the &quot;question&quot; bits.

In concrete for the example that the question describe:

a NN to classify dog/cat from 32x32 image. Two outputs for probabilities of dog an cat. Or simply one binary output 0:&quot;is a dog&quot;, 1:&quot;is a cat&quot;.
a second NN with the &quot;1025&quot; binary input ( 0: &quot;look for dogs&quot;, 1: &quot;look for cats&quot;) and output from previous one. In this case, if everything is ok, it will infer the logic &quot;a==b&quot;: &quot;is a dog&quot; and &quot;look for dogs&quot; or &quot;is a cat&quot; and &quot;look for cats&quot;.

Note that if you try to solve the problem directly with a full-connected NN with all inputs (1025 in the example), you loss the possibility of use CNN and max-pooling layers, etc. Moreover, split decreases training times, join subjects increases them &quot;exponentially&quot;. Not a promising way.